Abstract
Despite the apparently unrealistic assumption of infinite resources, infinite-server queueing models have played a central role in the development of queueing theory and its applications. Healthcare modelling applications have certainly benefited from these models, where arguably their greatest importance has been to provide the basis for the analysis of “offered load” in systems with single or multiple nodes with multiple servers and time-varying arrivals. In this paper, we provide a review of major healthcare applications to date, identifying and consolidating the underpinning theoretical results and commenting on the nature of the applications. We conclude by identifying potential further healthcare applications, their relationships to existing theory and methods, and the need for new theory and methods, including the use of infinite-server models alongside other modelling methodologies.
Highlights
The well-known result that the steady-state distribution of the number of customers in an M/G/1 queueing system is Poisson with mean equal to k=l; i.e., the ratio of the arrival rate ðkÞ to the service rate per server ðlÞ; is a classic example of an operational research model
There is an unlimited number of servers. Armed with this queueing model, operational researchers could perform “back-of-an-envelope” calculations to provide decision-makers with sound underestimates of the levels of congestion and variability to be expected in real systems that they were trying to manage; and an explanation of the extent to which the impact of this variability was likely to be less in bigger systems
Infinite-server queueing models have been developed in many directions since these early ideas
Summary
The well-known result that the steady-state distribution of the number of customers in an M/G/1 queueing system is Poisson with mean equal to k=l; i.e., the ratio of the arrival rate ðkÞ to the service rate per server ðlÞ; is a classic example of an operational research model. Whitt (2016), in reviewing work on infinite-server queues, comments on the central role that they have played in the development of queueing theory and applications, despite their assumption of infinite resource and no queues Alongside their importance in understanding many of the dynamics of time-dependent queues and the development of asymptotic results for “many server queues”, Whitt notes that arguably their greatest importance is to provide the basis for the analysis of “offered load” for multi-server systems with time-varying arrivals.
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